On MIMO Channel Capacity with Output Quantization Constraints

Transcription

1 On MIMO Channel Capacty wth Output Quantzaton Constrants Abbas Khall NYU New York, USA Stefano Rn NCTU Hsnchu, Tawan Luca Barletta Poltecnco d Mlano Mlano, Italy luca.barletta@polm.t Elza Erkp NYU New York, USA elza@nyu.edu Yonna C. Eldar Technon Hafa, Israel yonna@ee.technon.ac.l arxv: v1 [cs.it] 5 Jun 018 Abstract The capacty of a Multple-Input Multple-Output (MIMO) channel n whch the antenna outputs are processed by an analog lnear combnng network and quantzed by a set of threshold quantzers s studed. The lnear combnng weghts and quantzaton thresholds are selected from a set of possble confguratons as a functon of the channel matrx. The possble confguratons of the combnng network model specfc analog recever archtectures, such as sngle antenna selecton, sgn quantzaton of the antenna outputs or lnear processng of the outputs. An nterestng connecton between the capacty of ths channel and a constraned sphere packng problem n whch unt spheres are packed n a hyperplane arrangement s shown. From a hgh-level perspectve, ths follows from the fact that each threshold quantzer can be vewed as a hyperplane parttonng the transmtter sgnal space. Accordngly, the output of the set of quantzers corresponds to the possble regons nduced by the hyperplane arrangement correspondng to the channel realzaton and recever confguraton. Ths connecton provdes a number of mportant nsghts nto the desgn of quantzaton archtectures for MIMO recevers; for nstance, t shows that for a gven number of quantzers, choosng confguratons whch nduce a larger number of parttons can lead to hgher rates. 1 Index Terms MIMO, capacty, one-bt quantzaton, sphere packng, hybrd analog-dgtal recever. I. INTRODUCTION As the couplng of multple antennas and low-resoluton quantzaton hold the promse of enablng mllmeter-wave communcaton, the effect of fnte-precson output quantzaton on the performance of MIMO systems has been wdely nvestgated n recent lterature. In [1], the authors propose a general framework to study the capacty of MIMO channels wth varous output quantzaton constrants and derve some ntal results on the scalng of capacty n the number of avalable quantzaton levels. In ths paper, we further our understandng of output quantzaton constrants n MIMO channels by drawng a connecton between a constraned sphere packng problem and formulaton n [1]. Ths connecton suggest a rather nsghtful geometrc-combnatoral approach to the desgn of recever quantzaton strateges for MIMO channel wth output quantzaton. Lterature Revew: In [], low resoluton output quantzaton for MIMO channels s nvestgated through numercal evaluatons. The authors are perhaps the frst to note that the loss due to quantzaton can be relatvely small. Quantzaton 1 Ths work has been supported n part by NSF Grant # for the SISO channel s studed n detal where t s shown that, f the output s quantzed usng M bts, then the optmal nput dstrbuton need not have more than M 1 ponts n ts support. A cuttng-plane algorthm s employed to compute ths capacty and to generate optmum nput support. In [3], the authors study the capacty of MIMO channels wth sgn quantzaton of the outputs and reveal a connecton between a geometrc-combnatoral problem and the capacty of ths model at hgh SNR. Contrbutons: In the model of [1], the output of a MIMO channel s processed by an analog combnng network before beng fed to N tq threshold quantzers. The combnng network s chosen among a set of possble confguratons as a functon of the channel matrx: these confguratons represent analog operatons that can be performed by the recever analog frontend. Through the problem formulaton n [1], t s possble to study the performance of dfferent recever archtectures as a functon of the avalable quantzaton bts N tq and transmt power. In ths paper, we show that the capacty of the model n [1] can be approxmately characterzed usng the soluton of a geometrc-combnatoral problem. Each threshold quantzer n effect obtans a lnear combnaton of the nosy channel nputs and can thus be vewed as parttonng the transmt sgnal space wth a hyperplane. The output of the set of quantzers dentfes a regon among those nduced by the hyperplane arrangement correspondng to the channel matrx and recever confguraton. Transmtted ponts can be relably dstngushed at the recever when they are separated by a hyperplane n the transmt space. Our result generalzes those of [1], [3] and provdes an ntutve approach to desgn effectve, and sometmes surprsng, quantzaton strateges. For example, one would expect that, for a recever able to perform lnear combnaton before quantzaton, the optmal transmsson strategy s to perform Sngular Value Decomposton (SVD) followed by multlevel quantzaton of each sub-channel. We show that ths scheme s actually sub-optmal at hgh SNR as recever confguratons whch nduce a larger number of parttons may lead to hgher transmsson rates. Organzaton: The channel model s ntroduced n Sec. II. Combnatoral notons are presented n Sec. III. Pror results and a motvatng example gven n Sec. IV. The man result s presented n Sec. V. Sec. VI concludes the paper.

2 T x X 1 X Channel H Z 1 Z Z 3 W 1 W W 3 Analog combner V Fg. 1: System model for N t =, N r = 3, and N tq = 4. Notaton: All logarthms are taken n base two. The vector dag{m} s the dagonal of the matrx M whle λ(m) s the vector of egenvalues of M. The dentty matrx of sze n s I n, the all zero/all one matrx of sze n m s 0 n m /1 n m, respectvely. Dmensons for these matrces are omtted( when ) n mpled by the context. We adopt the conve1nton that = 0 for > n. II. CHANNEL MODEL Consder the dscrete-tme MIMO channel wth N t /N r transmt/receve antennas n whch an nput vector X n = [X 1,n... X Nt,n] T results n the output vector W n = [W 1,n... W Nr,n] T accordng to the relatonshp W n = HX n Z n, 1 n N, (1) where Z n s an..d. Gaussan vector of sze N r wth zero mean and unt varance and H s a full rank matrx of sze N r N t (.e. rank(h) = mn(n t, N r )), fxed through the transmsson block-length and known at the recever and transmtter. The nput s subject to the power constrant N n=1 E[ X n ] NP where X n s the -norm. We study a varaton of the model n (1) shown n Fg.1 n whch the output vector W n s processed by a recever analog front-end and fed to N tq threshold one-bt quantzers. Ths results n the vector Y n = [Y 1,n... Y Ntq,n] T { 1, 1} Ntq gven by Y n = sgn(vw n t), 1 n N, () where V s the analog combnng matrx of sze N tq N r, t s a threshold vector of length N tq and sgn(u) s the functon producng the sgn of each component of the vector u as plus or mnus one. The matrx V and the vector t are selected among a set of avalable confguratons F [1]: {V, t} F { R Ntq Nr, R Ntq}. (3) For a fxed recever confguraton, {V, t}, the capacty of the channel n () s obtaned as C(V, t) = t 1 t t 3 t 4 Y 1 Y Y 3 Y 4 max I(X; Y). (4) P X (x), E[ X ] P We are nterested n determnng the largest attanable performance over all possble confguratons, leadng to C(F) = max C(V, t). (5) {V,t} F The full rank assumpton s justfed for rchly scatterng envronments. In the followng, we provde an approxmate characterzaton of the soluton of the maxmzaton n (5) under the assumpton that dag{hh T } = dag{vv T } = 1 1 Nr. Under ths assumpton, the dervaton of the results s partcularly succnct and thus fttng to the avalable space. The more general case of any arbtrary channel matrx H and any combnng matrx V s presented n [4]. III. COMBINATORIAL INTERLUDE Ths secton brefly ntroduces a few combnatoral concepts useful n the remander of the paper. A hyperplane arrangement A s a fnte set of n affne hyperplanes n R m for some n, m N. A hyperplane arrangement A = {x R m, a T x = b } n =1 can be expressed as A = {x, Ax = b} where A s obtaned by lettng each row correspond to a T and defnng b = [b 1... b n ] T. A plane arrangement s sad to be n General Poston (GP) f and only f every n n sub-matrx of A has non zero determnant [5]. Lemma III.1. A hyperplane arrangement of sze n n R m dvdes R m nto at most r(m, n) = ( ) m n =0 n regons. Lemma III.. A hyperplane arrangement of sze n n R m where all the hyperplanes pass through the orgn dvdes R m nto at most r 0(n, m) = m ) =0 regons. ( n 1 Lemma III.3. Let A be a hyperplane arrangement of sze l n R m and consder a hyperplane arrangement B of sze dl wth d N hyperplanes parallel to each of the hyperplanes n A. Then B dvdes R m nto at most r p (m, n, d) = m ( l =0 ) d (1 d) l regons. A necessary condton to attan the qualtes n Lem. III.1, Lem. III. and Lem. III.3 s for the hyperplane arrangement A to be n GP. A untary sphere packng n R m s defned as P = N S m (c, 1), (6) =1 where S m (c, r) s the m-dmensonal hyper-sphere wth center c and radus r. A hyperplane separates two spheres f each sphere belongs to one of the half-spaces nduced by the hyperplane. A sphere packng P s sad to be separable by the hyperplane arrangement A f any two spheres are separated by at least one hyperplane n A. A sphere packng n a sphere s a packng P for whch P S(c, r) for some c, r. Our am s to show a connecton between the capacty n (5) and the followng sphere packng problem. Defnton III.4. Separable sphere packng n a sphere: Gven a hyperplane arrangement A and a constant r R, defne r ssps (A, r) as the largest number of unt spheres n a packng P contaned n the sphere S(0, r) and separable by A. IV. PRIOR RESULTS AND A MOTIVATING EXAMPLE The maxmzaton n (5) yelds the optmal performance for any set of possble recever confguratons. One s often nterested n studyng and comparng the performance for specfc classes of recever confguratons: three such classes

3 are studed n detal n [1]: sngle antenna selecton and multlevel quantzaton, sgn quantzaton of the outputs and lnear combnng and multlevel quantzaton. A. Pror Results The smplest recever archtecture of nterest s perhaps the one n whch a sngle antenna output s selected by the recever and quantzed through a hgh-resoluton quantzer. Ths s obtaned n the model of Sec. II by settng F = { V = [ 0 Ntq 1 N tq 1 0 Ntq N r 1 ], 0 N r 1 t R Ntq}, (7) where the term 1 Ntq 1 selects the antenna wth the hghest channel gan. Proposton 1. [1, Prop. ]. The capacty of the MIMO channel wth sngle antenna selecton and multlevel quantzaton s upper bounded as C select 1 log ( mn { 1 h T max P, (N tq 1) }), (8) where h T max s the row of H wth the largest -norm. The upper bound n (8) can be attaned to wthn bts-per-channel-use (bpcu). Our man result, dscussed n detal n Sec.V, s nspred by an ntrgung connecton between Lem. III. and the nfnte SNR capacty of the MIMO channel wth sgn quantzaton of the outputs [3]. Note that the model n [3] s obtaned by settng N tq = N r and lettng F be the set of all matrces obtaned by permutng the rows of [I, 0]. Proposton. [3, Prop. 3]. The capacty of the MIMO channel wth sgn quantzaton of the outputs n whch H s n GP at nfnte SNR s bounded as log(r 0 (N r, N t )) C SNR sgn log(r 0 (N r, N t ) 1). Recall that the most general archtecture n Sec. II has F = { V R Ntq Nr, t R Ntq}, (9) and corresponds to a recever analog front-end whch s able to perform any lnear combnaton of the antenna outputs before quantzaton. Proposton 3. [1, Prop. 6]. The capacty of a MIMO channel wth lnear combnng and multlevel quantzaton s upper bounded as C lnear R (λ(h), P, N tq ) K. (10) The capacty s to wthn a gap of 3K bpcu from the upper bound n (10) for R (λ(h), P, N tq) = K 1 =1 log(1 λp) f K ( ) =1 1 λ P 1 N tq ( ) Ntq K log K 1 otherwse, (11) wth K = max{n t, N r }, P = (µ λ ) and µ R s the smallest value for whch P = P. To establsh the achevablty of Prop. 3, the SVD can be used to transform the channel nto K = mn{n t, N r } parallel sub-channels wth ndependent unt-varance addtve nose and gans λ(h). After SVD, the quantzaton strategy s chosen dependng on whether the performance s bounded by the effect of the addtve nose or by the quantzaton nose. B. Motvatng Example for the Combnatoral Approach Let us consder the three archtectures n Propostons 1-3 for the case of N t =, N r = 3 and N tq = 4, also shown n Fg. 1, and provde some hgh-level ntuton on the relatonshp between capacty and the sphere packng problem n Def. III.4. Prop. 1: Snce the threshold quantzers are used to sample the same antenna output, the number of possble outputs s at most N r 1 so that the performance n Prop. 1 s bounded by log(n tq 1) = log 5.3 bpcu at hgh SNR. Ths recever confguraton can be nterpreted as follow: an antenna output represents a lne n the two-dmensonal transmt sgnal space; each threshold quantzer corresponds to a translaton of ths lne and these N tq parallel lnes partton the sgnal space nto at most N tq 1 subregons. Prop. : Sgn quantzaton of the outputs corresponds to the hyperplane arrangement n whch all hyperplanes pass through the orgn: the number of regons nduced by ths arrangement s obtaned through Lem. III.. There are r 0 = 8 parttons, as also shown n Fg. b, yeldng a maxmum rate of 3 bpcu, attanable at hgh SNR. Prop. 3: When the recever can perform lnear combnng before quantzaton, the SVD can be used to transform the channel nto two parallel sub-channels. Ths strategy corresponds to the hyperplane arrangement n Lem. III.3 and the number of parttons nduced s 9, as also shown n Fg. c. Lem. III.1: Ths lemma actually ndcates that the largest number of regons s 11 so that the rate log(11) = 3.46 bpcu can be obtaned through the recever confguraton n Fg. d at hgh SNR. 3 Gven the above nterpretaton of the capacty at hgh SNR, a feasble fnte SNR strategy s the one n whch, for a gven recever confguraton, the channel nputs are chosen as the center of the spheres wth suffcently large radus nsde each partton subject to the power constrant. The average achevable rate of the four strateges dscussed above s plotted n Fg. 3. Each lne n Fg. 3 corresponds to one of the sphere packng confguratons n Fg.. For a gven channel realzaton, V and t are chosen to result n the parttonngs of the transmtter space correspondng to each of the subfgures n Fg., approprately scaled by the avalable transmt power. Note that the confguratons are not optmzed. The channel nputs are then chosen as unformly dstrbuted over the center of the spheres packed n the parttonngs. 3 Note that ths does not contradct the result of Prop. 3 snce the nner bound s bpcu from the outer bound.

4 (a) Confguraton correspondng to Prop.1 (b) Confguraton correspondng to Prop. (c) Confguraton correspondng to Prop.3 (d) Confguraton nspred by Lem. III.1 Fg. : Dfferent recever output quantzaton strateges. Rate (bpcu) Fg. a Fg. b Fg. c Fg. d Outer bound and output alphabets as X = Y = [0 : r(n t, N tq )] and let the channel transton probablty be determned by the channel nput support and the recever analog confguraton. Also, let us defne sgn (x) as { sgn x x < 1 (x) = sgn(x) x 1, and the set N m as N m = {n n, n n S m (n, 1), n N m }, (14) P (db) Fg. 3: Smulaton results for N tq = 4, N r = 3, and N t = dscussed n Sec. IV-B. The average performance s calculated over real..d. zeromean, untary varance Gaussan channel gans, further scaled to guarantee that each row has untary -norm. The capacty of the channel wthout quantzaton constrant s also provded as a reference. From Fg. 3 we see that, at hgh SNR, the best performance s attaned by the confguraton correspondng to Lem. III.1, snce at hgh SNR the performance s determned by the number of transmtted ponts. As the SNR decreases, confguratons wth less transmtted ponts perform better. V. MAIN RESULT Sec. IV-B provdes a geometrc-combnatoral nterpretaton of the capacty of the model n (1)-() for the recever archtectures consdered n [1]. The man result of the paper s to make such nterpretaton more rgorous and more general. Theorem 1. The capacty expresson n (5) when dag{hh T } = dag{vv T } = 1 1 Nr s upper bounded as for C(F) max A log r ssps (A, P ) 3 K 3, (1) A {x, VHx = t, (V, t) F}, (13) and K = max{n t, N r }. The capacty s wthn.5n t bpcu from the outer bound n (1). Proof: Only the converse proof s presented here whle the achevablty proof s provded n [4]. Let us choose the nput that s N m s composed of a set of ponts selected from the unt sphere around the nteger ponts n N m. Fnally, let Q N m(x) be the mappng whch assgns each pont n R m to the closest pont n N m and ( W N = HQ N N (X N ), Y N = sgn VW N t t E N = W N W N. Usng Fano s nequalty, we wrte N(R ɛ N ) I(Y N, E N ; X N ) I(Y N ; X N ) H(E N ) H(Z N ) ), (15a) = I(Y N ; X N ) NN r log 3, (15b) where, n (15a), we have used the fact that we can reconstruct Y N from Y N and the value of E N. In (15b), we used the fact that snce dag{hh T } = 1, components of H(X Q Nt m(x)) have support at most [ 1, 1]. The largest varance of a random varable wth fnte support s for the case n whch the probablty dstrbuton s evenly dstrbuted at the end ponts, so that Var[E ] 3/. Usng the Gaussan maxmzes entropy property, we obtan H(E ) 1/ log(πe3). From a hgh-level perspectve, (15) shows that the capacty of the channel n (1)-() s close to the capacty of the channel wth no addtve nose but n whch the nput s mapped to N m. Next, we show that restrctng the nput to a peak power constrant, nstead of an average power constrant, has a bounded effect on the capacty. Let us represent X n hyper-geometrc coordnates as X = φ X for φ S Nt (0, 1) and φ = 1 and defne X N as X = φ ( X mod ) P, 1 N (16)

5 where mod (x) ndcates the modulus operaton; n other words, X has the same drecton as X but ts modulus s folded over P. Accordngly, defne Ŵ N = HQ N N ( X ( ) N ), Ŷ N = sgn VŴN t, t and use these defntons to further bound the term I(Y N ; X N ) n (15b) as I(Y N ; X N ) I(ŶN, Y N ; X N, X N ) (17) = I(ŶN ; X N ) I(ŶN ; X N X N ) (18) I(Y N ; X N, X N ŶN ). Note that I(ŶN ; X N X N ) = 0 because of the Markov chan Ŷ N X N X N. For the term I(Y N ; X N, X N ŶN ) we wrte I(Y N ; X N, X N ŶN ) H(Y N ŶN ) H(H(X N X N )) H(X N X N ), (19a) (19b) (19c) where (19a) follows from the fact that Y N s a dscrete random varable, (19b) from the fact that X and X are also dscrete random varables, and (19c) from the fact that H s full rank by assumpton. Next, to bound the term H(X N X N ), we can wrte X X = φ ( X / ) P, (0) where / ndcates the quotent of the modulus operaton. The entropy of ths random varable can then be rewrtten as H(X N X N ) H(φ N ) H( X N / P ) N(N r 1) N max H( X / P ). It can be shown that H( X / P ) 3 bpcu: the proof follows from the fact that the power constrant can be volated only a fnte number of tmes, whch leads to the fact that X / P s concentrated around small nteger values. By combnng the bounds n (15) and (17) we obtan N(R ɛ N ) I(ŶN ; X N ) 3 NK 3N N max I(Ŷ; X) 3 NK 3N. (1) P X Let us now evaluate the mutual nformaton term I(ŶN ; X N ), Ŷ N s a determnstc functon of X N and can be nterpreted as the membershp functon ndcatng to whch partton of the hyperplane arrangement the nput belongs to. For ths reason, I(ŶN ; X N ) s maxmzed by choosng an nput support as the subset of N m n whch a sngle pont s contaned n each partton nduced by {VHx = t} and lettng the nput dstrbuton be unformly dstrbuted over ths set. As a fnal step of the proof, we note that the upper bound n (1) can be mnmzed over the choce of the set N m n (14). In other words, by varyng the choce of n n n (14), the ponts n N m are moved outsde the correspondng partton, thus tghtenng the bound n (1). Accordngly, unless a partton contans a unt ball centered around a value n R Nr, a value n n can be chosen so that N m does not contan a value n such partton. It then follows that I(ŶN ; X N ) log r ssps (A, P ) whch s the desred result. Remark V.1. Th. 1 extends the results n Sec. IV-A as t holds for any set of possble recever confguratons F n (3). The results n Sec. IV-A only hold when F has a specfc form as n (7) or (9). On the other hand, Th. 1 does not provde a closed-form characterzaton of capacty as t nvolves the soluton of a packng problem. In partcular, lettng F n (13) have the form of (7) or (9) does not mmedately recover the capacty characterzaton n Sec. IV-A as Th. 1 follows a dfferent approach than [1] to bound capacty. Remark V.. When consderng the model wth any H and V, the result n Th. 1 generalzes as follows. The channel model n () s reduced to model where V and H are such that dag{hh T } = dag{vv T } = 1 1 Nr by lettng the addtve nose Z n have a general covarance matrx. For a channel model under ths normalzaton, consder the addtve nose after combnng, Z n = VZ n : the varance of the th entry of Z n, Z,n, determnes the uncertanty n the output of the th quantzer, Y,n. Accordngly, the capacty s approxmatvely equal to the number of separable ponts whch can be ftted n the sphere of radus P such that each pont s at dstance at least (Var[ Z,n ]) 1/ from the th hyperplane. The complete dervaton can be found n [4]. VI. CONCLUSION In ths paper, the capacty of a MIMO channel wth output quantzaton constrants for recevers equpped wth analog combners and one-bt threshold quantzers s nvestgated. The connecton between the capacty of the system and a constraned sphere packng problem s showed by argung that the threshold quantzers can be nterpreted as hyperplanes parttonng the transmt sgnal space. Ths connecton reveals, for example, that the nfnte SNR capacty of a channel wth lnear combner s attaned by a recever confguraton whch parttons the transmt sgnal space n the largest number of regons. REFERENCES [1] S. Rn, L. Barletta, Y. C. Eldar, and E. Erkp, A general framework for MIMO recevers wth low-resoluton quantzaton, Proc. IEEE Inf. Theory Workshop, Nov [] J. Sngh, O. Dabeer, and U. Madhow, On the lmts of communcaton wth low-precson analog-to-dgtal converson at the recever, IEEE Trans. Commun., vol. 57, no. 1, pp , 009. [3] J. Mo and R. W. Heath, Capacty analyss of one-bt quantzed MIMO systems wth transmtter channel state nformaton, IEEE Trans. Sgnal Process., vol. 63, no. 0, pp , 015. [4] A. Khall, S. Rn, L. Barletta, Y. Eldar, and E. Erkp, A general framework for low-resoluton recevers for MIMO channels, under preparaton. [5] T. Cover, Geometrcal and statstcal propertes of systems of lnear nequaltes wth applcatons n pattern recognton, IEEE Trans. Electron. Comput., vol. EC-14, no.3, pp , Jun